Hausmann and Rodriguez classified spaces of isometry classes of planar n-gons according to their genetic code which is a collection of sets (called genes) containing n. Omitting the n yields what we call gees. We prove that, for a set of gees with largest gee of size  the topological complexity (TC) of the associated space of n-gons is either  or  if  We present evidence that suggests that it is very rare that the TC is not equal to  or

topological complexity, planar polygon spaces.